Costas G. Papadopoulos∗a , Damiano Tommasinia and Christopher Wevera,b†
a Institute
of Nuclear and Particle Physics, NCSR ‘Demokritos’, Agia Paraskevi, 15310, Greece
for Theoretical Particle Physics (TTP), Karlsruhe Institute of Technology,
Engesserstraße 7, D-76128 Karlsruhe, Germany & Institute for Nuclear Physics (IKP),
Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1, D-76344
Eggenstein-Leopoldshafen, Germany
E-mail: [email protected]
b Institute
In this talk we present the Simplified Differential Equations (SDE) Approach for Master Integrals
(MI). Combined with the integrand reduction method for two-loop amplitudes it can pave the road
for a fully automated NNLO calculation framework. Most recent achievements of the method,
including double-box and pentabox MI calculations, are highlighted.
12th International Symposium on Radiative Corrections (Radcor 2015) and LoopFest XIV (Radiative
Corrections for the LHC and Future Colliders)
15-19 June, 2015
UCLA Department of Physics Astronomy Los Angeles, USA
∗ Speaker.
† This
research was supported by the Research Funding Program ARISTEIA, HOCTools (co-financed by the European Union (European Social Fund ESF) and Greek national funds through the Operational Program "Education and
Lifelong Learning" of the National Strategic Reference Framework (NSRF)).
c Copyright owned by the author(s) under the terms of the Creative Commons
Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).
http://pos.sissa.it/
PoS(RADCOR2015)104
Simplified Differential Equations Approach for
Master Integrals
Simplified Differential Equations Approach for Master Integrals
Costas G. Papadopoulos
1. Introduction
N (l1 , l2 ; {pi }) min(n,8)
∆i i ...i (l1 , l2 ; {pi })
= ∑ ∑ 12 m
.
D1 D2 . . . Dn
Di1 Di2 . . . Dim
m=1 Sm;n
(1.1)
where an arbitrary contribution to the two-loop amplitude (left), can be reduced to a sum of terms
(right) of all partitions Sm;n , with up to eight denominators; l1 , l2 are the loop momenta, Di are
the inverse scalar Feynman propagators, N (l1 , l2 ; {pi }) is a general numerator polynomial and
∆i1 i2 ...im (l1 , l2 ; {pi }) are the residues of multivariate polynomial division. In addition R2 terms [5]
have to be studied at two loops in order to achieve a complete framework. Moreover, two remarks
are in order here. The first is that the basis of two-loop integrals does not include only scalar
integrals. It includes integrals that also have irreducible scalar products (ISP) as numerators (to
some power) that cannot be rewritten as existing denominators of the integral. In the one-loop
case these ISP are always spurious and integrate to zero, but for higher loops this does no longer
hold. The second remark is that if one is interested in constructing an integral basis, the set of
integrals that ends up with is not necessarily a minimal one: the integrals are not by default Master
Integrals. At two or more loops one can find them by using integration-by-parts (IBP) identities
[14, 15, 16, 17, 18].
Contrary to the MI at one-loop, which have been known for a long time already [19], a complete library of MI at two-loops is still missing. At the moment this seems to be the main obstacle
to obtain a fully automated NNLO calculation framework similar to the one-loop case, that will
satisfy the anticipated precision requirements at the LHC [20].
2. The Simplified Differential Equations Approach
In the last fifteen years, the calculation of virtual corrections has been revolutionized with the
advent of automated reduction techniques to MI [15, 14, 16, 21] and the development of systematic
solutions of differential equations [22, 23, 24, 25] satisfied by MI or the evaluation of their MellinBarnes representations [26, 27]. The differential equations approach (DE) has proven to be very
powerful in a large number of computations, including two-loop four-point functions with massless
and massive internal propagators. Within this framework, DE for the MI are derived, in terms of
kinematical invariants. Assume that one is interested in calculating an l−loop Feynman integral
2
PoS(RADCOR2015)104
With LHC delivering collisions at the highest energy achieved so far, 13 TeV, experiments are
providing measurements of physical observables with an unprecedented precision. In order to keep
up with the increasing experimental accuracy as more data is collected, more precise theoretical
predictions and higher loop calculations are required [1].
With the better understanding of the reduction of one-loop amplitudes to a set of Master Integrals (MI) based on unitarity methods [2, 3] and at the integrand level via the OPP method [4, 5],
one-loop calculations have been fully automated in many numerical tools [6, 7]. In the recent years,
a lot of progress has been made towards the extension of these reduction methods to the two-loop
order at the integral [8, 9]as well as the integrand [10, 11, 12, 13] level. The master equation at the
integrand level can be given schematically as follows [13]
Simplified Differential Equations Approach for Master Integrals
Costas G. Papadopoulos
with external momenta {p j } and internal propagators that are massless. Any l−loop Feynman
integral can be then written as
!
Z
l
d d kr
1
Ga1 ···an ({p j }, ε) =
∏ iπ d/2 Da1 · · · Dann , Di = (ci j k j + di j p j )2 , d = 4 − 2ε (2.1)
1
r=1
variables, defined on a case by case basis, so that the resulting DE can be solved in terms of
Goncharov Polylogarithms (GPs) [29, 30].The GPs are defined recursively as follows
G (an , . . . , a1 , x) =
Zx
dt
0
1
G (an−1 , . . . , a1 ,t)
t − an
(2.2)
n
z }| {
with the special cases, G (x) = 1 and G (0, . . . , 0, x) = n!1 logn (x). Usually boundary terms corresponding to the appropriate limits of the chosen parameters have to be calculated using for instance
expansion by regions techniques [31, 32].
SDE approach [33] is an attempt not only to simplify, but also to systematize, as much as
possible, the derivation of the appropriate system of DE satisfied by the MI. To this end the external
incoming momenta are parametrized linearly in terms of x as pi (x) = pi + (1 − x)qi , where the qi ’s
are a linear combination of the momenta {pi } such that ∑i qi = 0. If p2i = 0, the parameter x
captures the off-shell-ness of the external legs. The class of Feynman integrals in (2.1) are now
dependent on x through the external momenta:
!
Z
l
1
d d kr
Ga1 ···an ({si j }, ε; x) =
(2.3)
∏ iπ d/2 Da1 · · · Dann , Di = (ci j k j + di j p j (x))2 .
1
r=1
~ MI ({si j }, ε; x), which now deBy introducing the dimensionless parameter x, the vector of MI G
pends on x, satisfies
∂ ~ MI
~ MI ({si j }, ε; x)
G ({si j }, ε; x) = H({si j }, ε; x)G
(2.4)
∂x
a system of differential equations in one independent variable. Experience up to now shows that
this simple parametrization can be used universally to deal with up to six kinematical scales involved [33, 34, 35]. The expected benefit is that the integration of the DE naturally captures the
expressibility of MI in terms of GPs and more importantly makes the problem independent on the
number of kinematical scales (independent invariants) involved. Note that as x → 1, the original
configuration of the loop integrals (2.1) is reproduced, which eventually corresponds to a simpler
one with one scale less.
3
PoS(RADCOR2015)104
with matrices {ci j } and {di j } determined by the topology and the momentum flow of the graph,
and the denominators are defined in such a way that all scalar product invariants can be written
as a linear combination of them. The exponents ai are integers and may be negative in order to
accommodate irreducible numerators. Any integral Ga1 ···an can be written as a linear combination
of a finite subset of such integrals, called Master Integrals, with coefficients depending on the
independent scalar products, si j = pi · p j , and space-time dimension d, by the use of integration
~ MI ({si j }, ε), are differentiated
by parts identities [15, 14]. In the traditional DE method, the MI G
∂
with respect to pi · ∂ p j , and the resulting integrals are reduced by IBP to give a linear system of
~ MI ({si j }, ε) [22, 28]. The invariants, si j , are then parametrised in terms of dimensionless
DE for G
Simplified Differential Equations Approach for Master Integrals
Costas G. Papadopoulos
3. Master Integrals calculations
The SDE approach has been proven useful in calculations of MI at one and two loops.
3.1 One-loop pentagon at O (ε)
The one-loop pentagon presented in Fig.1 is given by
Z
G11111 =
dd k
1
d/2
iπ (−k2 ) − (k + x p1 )2 − (k + xp12 )2 − (k + p123 )2 − (k + p1234 )2
(3.1)
with = 0, i = 1, . . . , 4, the obvious notation pi... j = pi + . . . + p j and
= (−p5 = 0. The
DE for the 5-point MI involves the following set of MI: four four-point G11110 , G11101 , G11011 ,
G10111 , one three-point, G10110 and three two-point MI, G00110 , G10010 , G10100 . The two-point MI
are known in a closed form in ε. All boundary conditions have been calculated by the DE, as
described in [33]. The result for the one-mass pentagon up to order ε, is given by
1
cΓ
1
x −1−ε
x −1−ε 1 i
1−
G11111 (x) = 2
1−
1−
1−
∑ ε fi (3.2)
x s23 s34 s45
r1
r2
r1
r2
i=−2
p2i
p21234
)2
where the quantities r1 , r2 , cΓ and fi can be found in [33]. Taking the limit x → 1 from the previous
expression, we get the result for the on-shell pentagon up to order ε, that is given by
G11111 (1) =
1
cΓ
εi f 0
∑
s12 s23 s34 s45 s51 i=−2 i
(3.3)
where again the fi0 can be found in [33].
3.2 Full set of massless double-box MI with two off-shell legs
There are in total six families of MI whose members with the maximum amount of denominators are graphically shown in Fig.2 and Fig.3. Three of these, Fig.2, contain only planar MI and
will therefore be referred to as the planar families. They will be denoted by P12 , P13 and P23 [36]
and contain 31, 29 and 28 MI respectively. The other three families, shown in Fig.3, contain planar
MI with up to 6 denominators as well as non-planar MI with 6 and 7 denominators and will be
referred to as the non-planar families of MI. These non-planar families will be denoted by N12 , N13
4
PoS(RADCOR2015)104
Figure 1: The one-loop pentagon graph with one off-shell leg, G11111 . The labels refer to the denominators
in Eq.(3.1).
Costas G. Papadopoulos
Simplified Differential Equations Approach for Master Integrals
xp1
−p123
xp2
p123 − xp12
xp2
xp1
xp1
p123 − xp12
−p123
p123 − xp12
−p123
xp2
Figure 2: The parametrization of external momenta for the three planar double boxes of the families P12
(left), P13 (middle) and P23 (right) contributing to pair production at the LHC. All external momenta are
incoming.
−p123
xp2
p123 − xp12
xp2
xp1
p123 − xp12
−p123
xp1
−p123
xp2
p123 − xp12
Figure 3: The parametrization of external momenta for the three non-planar double boxes of the families
N12 (left), N13 (middle) and N34 (right) contributing to pair production at the LHC. All external momenta are
incoming.
and N34 [37] and contain 35, 43 and 51 MI respectively. We have used both FIRE [38] and Reduze
2 [39] to perform the IBP reduction to the MI.
For instance the integrals in the family P12 are given by
(x, s, ε)
GPa12
1 ···a9
:=
e2γE ε
×
d d k1 d d k2
1
2a
d/2
d/2
1
2a
iπ iπ k1 (k1 + xp1 ) 2 (k1 + xp12 )2a3 (k1 + p123 )2a4
1
,
2a5
2a
6
k2 (k2 − xp1 ) (k2 − xp12 )2a7 (k2 − p123 )2a8 (k1 + k2 )2a9
Z
(3.4)
where the letters appearing as weights in GPs are given by
q s12
q
s23
s23
s12
I(P12 ) = 0, 1, , ,
,1−
,1+
,
s12 q q − s23
q
s12 s12 + s23
with s12 := p212 , s23 := p223 , q := p2123 , p2i = 0. Details of the calculations as well as the results in
terms of GPs for all families can be found in [34].
In order to compute the MI in arbitrary kinematics, especially in the physical region, the GPs
have to be properly analytically continued. In general all variables, including the momenta invariants si j (s12 , s23 and q in the present study) and the parameter x, would acquire an infinitesimal
imaginary part, si j → si j + iδsi j η, x → x + iδx η, with η → 0. The parameters δsi j and δx are determined as follows: the first class of constraints on the above-mentioned parameters originates
form the input data to the DE, namely the one-scale master integrals that need to be properly defined in each kinematical region. The second class of constraints results from the second graph
polynomial [40], F , which after expressed in terms of si j and x, should acquire a definite-negative
imaginary part in the limit η → 0. Combining these two classes of constraints on the parameters
δsi j and δx , the imaginary part of all the GPs involved is fixed and we have checked that the result
for the MI is identical in the limit η → 0 and moreover agrees with the one obtained by other
calculations. We have performed several numerical checks of all our calculations. The numerical
results have been compared with those provided by the numerical code SecDec [41, 42, 43] in the
5
PoS(RADCOR2015)104
xp1
Simplified Differential Equations Approach for Master Integrals
Costas G. Papadopoulos
Figure 4: The three planar pentaboxes of the families P1 (left), P2 (middle) and P3 (right) with one external
massive leg.
Euclidean region and with analytic results presented in [36, 37] for the physical region. In all cases
we find perfect agreement and reference numerical results can be found in the ancillary files in
[34].
3.3 Massless pentabox MI with up to one off-shell leg
For the massless pentabox MI with one off-shell leg, there are in total three families of planar
MI whose members with the maximum amount of denominators, namely eight, are graphically
shown in Fig.4. Similarly, there are five non-planar families of MI as given in Fig.5. We have
checked that the other five-point integrals with one massive external leg are reducible to MI in one
of these eight MI families. The two-loop planar and non-planar diagrams have not been calculated
yet; in [35] we have recently completed the calculation of the P1 family (Fig.4). In fact by taking the
limit x → 1 all planar graphs for massless on-shell external momenta have been derived as well 1 .
We have used the c++ implementation of the program FIRE [45] to perform the IBP reduction to
the set of MI in P1 .
For the family of integrals P1 the external momenta are parametrized in x as shown in Fig.6.
The parametrization is chosen such that the double box MI with two massive external legs that is
contained in the family P1 has exactly the same parametrization as that one chosen in [34], i.e. two
massless external momenta xp1 and xp2 and two massive external momenta p123 − xp12 and −p123 .
The MI in the family P1 are therefore a function of a parameter x and the following five invariants:
s12 := p212 , s23 := p223 , s34 := p234 , s45 := p245 = p2123 , s51 := p215 = p2234 , p2i = 0, where the notation
pi··· j = pi + · · · + p j is used and p5 := −p1234 . As the parameter x → 1, the external momentum
q3 := p123 − xp12 becomes massless, such that our parametrization also captures the on-shell case.
The resulting differential equation in matrix form can be written as
∂x G = M
1 Results
si j , ε, x G
related to massless planar pentabox appear also recently in [44].
6
(3.5)
PoS(RADCOR2015)104
Figure 5: The five non-planar families with one external massive leg.
Simplified Differential Equations Approach for Master Integrals
xp1
Costas G. Papadopoulos
−p1234
p4
p123 − xp12
xp2
Figure 6: The parametrization of external momenta in terms of x for the planar pentabox of the family P1 .
All external momenta are incoming.
where the residue matrices Mi are independent of x and ε. The result can be straightforwardly
given as
(−2)
(−2)
(−1)
(−2)
(−1)
(0)
G = ε −2 b0 + ε −1 ∑ Ga Ma b0 + b0
+ ∑ Gab Ma Mb b0 + ∑ Ga Ma b0 + b0
(3.7)
(−2)
(−1)
(0)
(1)
+ ε ∑ Gabc Ma Mb Mc b0 + ∑ Gab Ma Mb b0 + ∑ Ga Ma b0 +b0
(−2)
(−1)
(0)
(1)
(2)
+ ε 2 ∑ Gabcd Ma Mb Mc Md b0 + ∑ Gabc Ma Mb Mc b0 + ∑ Gab Ma Mb b0 + ∑ Ga Ma b0 +b0
(k)
with the arrays b0 , k = −2, ..., 2 representing the x-independent boundary terms in the limit x = 0 at order ε k .
The expression is in terms of Goncharov polylogarithms, Gab... = G (la , lb , . . . ; x). Details on the calculation
of boundary terms and of the x → 1 limit can be found in [35]2 .
The solution for all 74 MI contains O(3, 000) GPs which is roughly six times more than the corresponding double-box with two off-shell legs planar MI. We have performed several numerical checks of all
our calculations. The numerical results, also included in the ancillary files [47], have been performed with
the GiNaC library [48] and compared with those provided by the numerical code SecDec [41, 42, 43, 49]
in the Euclidean region for all MI and in the physical region whenever possible (due to CPU time limitations in using SecDec) and found perfect agreement. For the physical region we are using the analytic
continuation as described in the previous section as well as in [34]. At the present stage we are not setting a
fully-fledged numerical implementation, which will be done when all families will be computed. Our experience with double-box computations show that using for instance HyperInt [50] to bring all GPs in their
range of convergence, before evaluating them numerically by GiNaC, increases efficiency by two orders
of magnitude. Moreover expressing GPs in terms of classical polylogarithms and Li2,2 , could also reduce
substantially the CPU time [51]. Based on the above we estimate that a target of O 102 − 103 milliseconds
can be achieved.
4. Outlook
We have demonstrated that based on the Simplified Differential Equations approach [33] Master Integrals, including massless double-box with two off-shell legs and pentabox with up to one off-shell leg, can
2 See
also talk by C. Wever.
7
PoS(RADCOR2015)104
where G stands for the array of the 74 MI involved in the family P1 . The twenty letters li involved,
are given in [35]. Although the DE can be solved starting from , e.g. (3.5), and the result can be
expressed as a sum of GPs with argument x and weights given by the letters li , it is more elegant
and easy-to-solve to derive a Fuchsian system of equations [46], where only single poles in the
variable x will appear. A series of transformation described in [35], brings the system into the form
!
19
Mi
∂x G = ε ∑
G
(3.6)
i=1 (x − li )
Simplified Differential Equations Approach for Master Integrals
Costas G. Papadopoulos
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